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Tukey-Lambda Distribution

NIST/SEMATECH Section 1.3.6.6.15 Tukey-Lambda Distribution

PDF

TL(l=0.14) PDF -4 -3 -2 -1 0 1 2 3 4 x 0 0.05 0.1 0.15 0.2 0.25 0.3 f(x)

CDF

TL(l=0.14) CDF -4 -3 -2 -1 0 1 2 3 4 x 0 0.2 0.4 0.6 0.8 1 F(x)

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PDF

CDF

Formulas

Probability Density Function

f(x)=1Q′(F),Q(F)=Fλ−(1−F)λλf(x) = \frac{1}{Q'(F)}, \quad Q(F) = \frac{F^\lambda - (1-F)^\lambda}{\lambda}

Cumulative Distribution Function

F(x)=Q−1(x),Q(F)=Fλ−(1−F)λλF(x) = Q^{-1}(x), \quad Q(F) = \frac{F^\lambda - (1-F)^\lambda}{\lambda}

Properties

Mean

00

Variance

2λ2(11+2λ−Γ(λ+1)2Γ(2λ+2))\frac{2}{\lambda^2}\left(\frac{1}{1+2\lambda} - \frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\right)

Overview

The Tukey-Lambda distribution is a symmetric family defined by its quantile function. By varying the shape parameter λ\lambda, it can approximate the normal, logistic, Cauchy, and uniform distributions.

Related Distributions

Normal Distribution Cauchy Distribution Uniform Distribution